206 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



Moreover, if we write ; , , ^/v 7 / for the differential coefficients 



da dp dy 



obtained from (420) by treating a, ft', y as independent variables, 



when 



and a'=l, /3' = 0, y ' = 



That is, ' 



when a' = l, /3' = 0, y' = 0. 



Hence, ^ = 0, = 0. , 



Therefore a line of the element which in the unstrained state is per- 

 pendicular to X' is perpendicular to X in the strained state. Of all 

 such lines we may choose one for which the value of r is at least as 

 great as for any other, and make the axes of Y' and Y parallel to this 

 line in the unstrained and in the strained state respectively. Then 



0; ' (424) 



and it may easily be shown by reasoning similar to that which lias 

 just been employed that 



Lines parallel to the axes of X', Y', and Z' in the unstrained body 

 will therefore be parallel to X, F, and Z in the strained body, and the 

 ratios of elongation for such lines will be 



dx dy dz 

 dx" dy" US' 



These lines have the common property of a stationary value of the 

 ratio of elongation for varying directions of the line. This appears 

 from the form to which the general value of r 2 is reduced by the 

 positions of the co-ordinate axes, viz., 



Having thus proved the existence of lines, with reference to any 

 particular strain, which have the properties mentioned, let us 

 proceed to find the relations between the ratios of elongation 

 for these lines (the principal axes of strain) and the quantities 



